#7509523

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,32)(21,35)(22,25)(23,27)(26,34)(28,39)(29,44)(30,37)(31,40)(33,41)(36,42)(38,45)(43,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,31)(23,28)(25,40)(26,38)(27,39)(32,44)(33,43)(34,45)(35,37)(36,46)(41,47)(42,48);(9,12,13,11)(10,14,15,16)(18,25)(19,28)(21,23)(24,37)(26,32)(29,43)(30,31)(33,36)(38,47)(39,40)(44,48)(45,46);(2,7)(3,8)(10,15)(14,16)(17,20,22,34)(18,26,25,32)(19,29,31,45)(21,36,23,33)(24,38,40,44)(27,42,35,41)(28,43,30,46)(37,47,39,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,25,35)(19,30,31,28)(20,33,34,36)(24,39,40,37)(26,41,32,42)(29,43,45,46)(38,47,44,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416719);